What's nonsensical about this definition of the order of a meromorphic function?

According to this math.stackexchange.com answer, the following definition of Huybrechts in his book Complex Geometry is nonsensical:

Let $X$ be a complex manifold. Let $Y subset X$ be a hypersurface and let $x in Y$. Suppose that $Y$ defines an irreducible germ in $x$. Hence, this germ is the zero set of an irreducible $g in mathcal{O}_{X,x}$.

Definition. (D. Huybrechts, Complex Geometry, Definition 2.3.5, page 78) Let $f$ be a meromorphic function in a neighbourhood of $x in Y$. Then, the order $mathrm{ord}_{Y,x}(f)$ of $f$ in $x$ with respect to $Y$ is given by the equality $f = g^{mathrm{ord}(f)}cdot h$ with $h in mathcal{O}^*_{X,x}$.

The definition seems reasonable to me. What’s wrong with it?

For example, suppose $X = mathbb{C}^2$, $Y = {0}timesmathbb{C}$, and $f(x, y) = frac{1}{x} + y$. Then, $g(x,y) = x$, and we have $f = g^{-1}h$ where $h(x,y)=1+xy in mathcal{O}_{X,(0,0)}^*$. So, according to Huybrechts’ definition, we have $mathrm{ord}_{Y,(0,0)}(f)=-1$, which seems correct to me.